| L(s) = 1 | + 2·3-s − 3·4-s + 9-s − 6·12-s + 4·16-s − 14·17-s − 12·23-s + 18·25-s − 2·27-s + 2·29-s − 3·36-s + 12·43-s + 8·48-s − 10·49-s − 28·51-s − 36·53-s − 2·61-s − 9·64-s + 42·68-s − 24·69-s + 36·75-s − 16·79-s − 4·81-s + 4·87-s + 36·92-s − 54·100-s + 6·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s + 1/3·9-s − 1.73·12-s + 16-s − 3.39·17-s − 2.50·23-s + 18/5·25-s − 0.384·27-s + 0.371·29-s − 1/2·36-s + 1.82·43-s + 1.15·48-s − 1.42·49-s − 3.92·51-s − 4.94·53-s − 0.256·61-s − 9/8·64-s + 5.09·68-s − 2.88·69-s + 4.15·75-s − 1.80·79-s − 4/9·81-s + 0.428·87-s + 3.75·92-s − 5.39·100-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2484383424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2484383424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 18 T^{2} + 203 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 2 T^{2} - 357 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 + T^{2} - 1680 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 18 T^{2} - 7597 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.944254099659408466843768717945, −7.72990013625310913527757562497, −7.67527142596322194971625883130, −7.10852741459749252688015635459, −6.92289583815323643718417550218, −6.70033219465552563070497007774, −6.34018440805261353762613926376, −6.23463890766177822734991122110, −6.09692496630906208277571074719, −5.70069964516609289529625809262, −5.07683216000238091358954193731, −5.04669575786185700694298464037, −4.69577444704819325306802975315, −4.56660881705694473245336617326, −4.25679356984428920456548589490, −4.23561651004958191088107881738, −3.78992274906571636704038272722, −3.34110204383868907799501070667, −3.00957761668490663918741288685, −2.84200221616373123076709201883, −2.53724780255686030195816493210, −1.93980892458886196111240488829, −1.81578555067958701601834462724, −1.15376585142794175515460491807, −0.15960876978430283758521300916,
0.15960876978430283758521300916, 1.15376585142794175515460491807, 1.81578555067958701601834462724, 1.93980892458886196111240488829, 2.53724780255686030195816493210, 2.84200221616373123076709201883, 3.00957761668490663918741288685, 3.34110204383868907799501070667, 3.78992274906571636704038272722, 4.23561651004958191088107881738, 4.25679356984428920456548589490, 4.56660881705694473245336617326, 4.69577444704819325306802975315, 5.04669575786185700694298464037, 5.07683216000238091358954193731, 5.70069964516609289529625809262, 6.09692496630906208277571074719, 6.23463890766177822734991122110, 6.34018440805261353762613926376, 6.70033219465552563070497007774, 6.92289583815323643718417550218, 7.10852741459749252688015635459, 7.67527142596322194971625883130, 7.72990013625310913527757562497, 7.944254099659408466843768717945
